Obtaining liquids of precisely known composition is in many cases important, as when buffers having a specified pH and optionally also ionic strength are utilised. Further, in many cases, the composition of the liquid should not only be at each moment precisely known and controlled, but also should vary with time in a precise and controlled manner.
One application where the composition of liquids is of utmost importance is in liquid chromatography, and more specifically when elution i.e. the release of isolated target molecules from the chromatography matrix, is carried out by gradient elution. For example, in ion exchange chromatography, which is a frequently used method for the separation and purification of biomolecules, gradient elution is sometimes used, e.g. for finding the optimal elution conditions enabling the design of an industrial process utilizing step-wise elution. As is well known, the eluent then contains an inert salt and the gradient is performed by varying the concentration of this salt. It is well known that a change in salt concentration i.e. ionic strength also affects the pH, and it has been well documented that the pH and ionic strength of the eluent are the two most important parameters that control selectivity of protein separations on ion exchange resins.
U.S. Pat. No. 5,112,949 (Vukovich) relates to an automated system for performing a separation using a gradient. A problem with such manual and automated systems with gradually changing gradients is that if the gradient is made shallow, then it takes a lot of time to perform the elution and if the gradient is made steep then instead of each biomolecule of interest being eluted in turn the elution of the biomolecules overlap. This leads to several species of biomolecules being collected in each fraction instead of each specie of biomolecule being collected in its own, separate fraction.
U.S. Pat. No. 7,138,051 (De Lamotte) relates to a chromatography system, method and software for the separation of biomolecules. More specifically, this patent relates to optimization of the separation of biomolecules eluted from a chromatography column in which the concentration of a component added to an elution buffer is varied in order to form an elution buffer solution of gradually changing concentration of said added component.
Okamoto (Hirokazu Okamoto et al, Pharmaceutical Research. Vol. 14, No. 3, 1997: Theory and Computer Programs for Calculating Solution pH, Buffer formula, and Buffer Capacity for Multiple Component System at a Given Ionic Strength and Temperature) described computer programs to calculate solution pH, buffer formula and buffer capacity at a given ionic strength and temperature. However, the buffer solutions prepared by Okamoto et al show a pH variation that increase with the increased salt concentration, even at low concentrations such as 0.3 M. Thus, the Okamoto methods are not sufficient to provide buffer solutions of constant pH which also contain salt.
The traditional way of gradient formation has involved the careful preparation of eluents comprising inert salts as well as buffers of predetermined pH to effect the ionic strength gradient at constant pH. The optimization of the separation of the proteins has been accomplished by changing the slope of the inert salt gradient and/or replacing the buffer system by one with a different pH.
In the early prior art, the optimization included the preparation of numerous buffer solutions with predetermined pH and salt concentrations, which had to be meticulously titrated for the separations to be reproducible. Obviously, such methods are both time consuming and awkward.
Methods for the calculation of buffer pH at moderate ionic strengths (up to 100 mM) are documented in the literature and are based on the algebraic or computer based solution of the equations of equilibrium among the various charged and uncharged species present in the buffer solution.
For a particular basic species (which can be a base B or a conjugate base A−) in equilibrium with a corresponding acidic species (which can be a conjugate acid BH+ or an acid HA, respectively) the equilibrium can be writtenH++basic species<==>acidic species+  Eq 1.1
The corresponding equilibrium constant Ka is defined asKa=(H+)(basic species)/(acidic species+)  Eq 1.2wherein the parenthesis denotes the activities of each species. Taking the logarithms of both side of Eq. 1.2 and solving for the pH defined as −log(H+) givespH=pKa+log {(basic species)/(acidic species)}  Eq 1.3which is sometimes known as the Henderson-Hasselbach equation. The reason why the activities are to be used in Eq 1.2 rather than the corresponding concentrations is that due to mainly electrostatic interactions, the ions involved tend to become shielded from the environment. However, whereas pH measurements are direct observations of the activity of the protons, it is rather the concentrations and not the corresponding activities of the buffer ions which are observed for instance by weighting, pippeting or pumping their amounts and volumes. The activity of each ion is related to the corresponding concentration through the activity coefficient φ(species)=φ[species]  Eq 1.4
At the ideal state of infinite dilution, φ becomes 1 and the activity of every ion become equal to the corresponding concentration. However, in real cases, the ionic strength is different from 0 and the activity coefficients of the different species become less than 1.
A well established model for these deviations has been developed in the so called Debye Hückel theory, known as−log φ=(AZ2I0.5)/(1+0.3*108aI0.5)  Eq 1.5wherein A is a constant, or rather a temperature dependent parameter ˜0.51. Using well known data, the value of A can accurately be calculated as A=0.4918+0.0007*T+0.000004*T^2 where T is the temperature in degrees Celsius. Z is the charge of the ion and the quantity a, the radii of the hydrated ions (in Å), is described as the “mean distance of approach of the ions, positive or negative” in the original paper of Debye and Hückel.
In a table presented in the above-discussed article by Kielland, this parameter, also known as the ion size parameter is shown to be different for different ionic species. I is the ionic strengthI=½Σ(CiZi2) (includes all ions)  Eq 1.6Ci is the concentration and Zi is the charge of ion present in the solution (in units of electronic charge).
Inserting Eq 1.4 into Eq 1.3 gives the pH in terms of the concentrations instead of the activities:
                                                                        pH                =                                ⁢                                                      pK                    a                                    +                                      log                    ⁢                                          {                                                                                                    φ                            b                                                    ⁡                                                      [                                                          basic                              ⁢                                                                                                                          ⁢                              species                                                        ]                                                                          /                                                  (                                                                                    φ                              a                                                        ⁡                                                          [                                                              acidic                                ⁢                                                                                                                                  ⁢                                species                                                            ]                                                                                }                                                                                                                                                                                            =                                ⁢                                                      pK                    a                                    +                                      log                    ⁢                                                                                  ⁢                                          φ                      b                                                        -                                      log                    ⁢                                                                                  ⁢                                          φ                      a                                                        +                                      log                    ⁢                                          {                                                                        [                                                      basic                            ⁢                                                                                                                  ⁢                            species                                                    ]                                                /                                                                                                                                                                                    ⁢                                  [                                      acidic                    ⁢                                                                                  ⁢                    species                                    ]                                }                                                                                        =                                ⁢                                                      pK                    a                    ′                                    +                                      log                    ⁢                                          {                                                                        [                                                      basic                            ⁢                                                                                                                  ⁢                            species                                                    ]                                                /                                                  [                                                      acidic                            ⁢                                                                                                                  ⁢                            species                                                    ]                                                                    }                                                                                                          ⁢                                  ⁢        where                            Eq        .                                  ⁢        1.7                                          pK          a          ′                =                              pK            a                    +                      log            ⁢                                                  ⁢                          φ              b                                -                      log            ⁢                                                  ⁢                          φ              a                                                          Eq        .                                  ⁢        1.8            is an apparent pKa value which allows the use of the measurable values of the concentrations of the different buffer species. The value of pKa′ can be calculated inserting Eq 1.5 into Eq 1.8 givingpKa′=pKa+(AZa2I0.5)/(1+0.33*108aaI0.5)−(AZb2I0.5)/(1+0.33*108abI0.5)  Eq 1.9where the introduction of the subscripts a and b was necessary to specify the parameters corresponding to the acid and the base respectively. ThusZa=Charge of acidic speciesZb=Charge of basic speciesaa=ion size parameter of the acidic speciesab=ion size parameter of the acidic species
Applied to pH calculations, the Debye-Hückel theory results in the modification of the pKa values of the buffers (known as the thermodynamic pKa values) into corresponding pKa′ values given by Eq 1.9. Most of the parameters in Eq 1.9 are straight forward to estimate. The most challenging parameter is a.
Guggenheim & Schindler (see Guggenheim E A & Schindler T D. (1934) J. Phys. Chem. 33. 533), has suggested an approximation of the parameter a set to 3 Å for all buffer molecules leading to the somewhat simplified formulapKa′=pKa+(AZa2I0.5)/(1+I0.5)−(AZb2I0.5)/(1+I0.5)  Eq 1.10Eq 1.10 above is the formula for ionic strength correction usually found in the literature. Sometimes correction terms are added to the right hand side of this equation to compensate for accuracy loss at higher ionic strengths for various buffers. However, the accuracy obtained by doing this is poor when the ionic strength is as high as 1M, which is within commonly used ranges in gradient elution in for instance ion exchange chromatography and HIC.
Kielland (Jacob Kielland in Activity Coefficients of Ions in Aqueous Solutions, September 1937) has studied activity coefficients of ions in liquids and provides an extended table of ionic activity coefficients, taking into consideration the diameter of the hydrated ions. The data presented by Kielland for the hydrated ion size parameter ai was obtained using four different models: Bonino's model which takes into account the crystal radius and deformability; the well known equation 108ai=182zi/1∞, which takes ionic mobilities into consideration; the empirical modification thereof by Brull, and finally the Ulrich entropy deficiency method. Rounded average values of said four models were used to obtain the data reported in that study. The ai values presented by Kielland present a substantial variation, from 2.5 to as much as 11, and non-general models are suggested for the activity coefficient based on this variation dependent upon the nature of the ions i.e. one equation for inorganic ions and one different equation for organic ions.
U.S. Pat. No. 6,221,250 (Stafström) relates to a method of preparing liquid mixtures which advantageously utilizes the above-discussed approximation of the parameter a. More specifically, the disclosed method of preparing a mixture comprises the following components: (i) one or more buffering species; (ii) an acid or alternatively a base; (iii) optionally a salt; and (iv) a solvent. The proportions of the components (i) to (iv) are concomitantly varied in such a way as to take account of the interrelationship of the pH and the ionic strength of the liquid mixture to obtain at each moment a preselected pH of the mixture, and the method is based on the use of a modified and repetitive Guggenheim-Schindler (1.10, below) equation wherein buffer specific correction factors are used for attainment of constant pH along a gradient. Thus, in certain situations, a disadvantage of this method can be that if a new buffer needs to be introduced; calculations need to be made again.
Another application where it is essential to prepare liquid mixtures of controlled pH and ionic strength is in high throughput screening (HTS), which is a method for scientific experimentation frequently used in drug discovery but also relevant to the fields of biology and chemistry. Through a combination of modern robotics, data processing and control software, liquid handling devices and sensitive detectors, HTS allows a researcher to effectively conduct millions of biochemical, genetic or pharmacological tests within a short period of time. Through this process one can rapidly identify active compounds, antibodies or genes which modulate a particular biomolecular pathway. The results of these experiments may e.g. provide starting points for drug design and for understanding the interaction or role of a particular biochemical process in biology. Automation is an important element in HTS's usefulness. A specialized robot is often responsible for much of the process over the lifetime of a single assay plate, from creation through final analysis. An HTS robot can usually prepare and analyze many plates simultaneously, further speeding the data-collection process. However, for these robots to function accurately, again the preparation of liquid mixtures such as buffers having precisely controlled pH as well as ionic strength is essential.
One application within the HTS area which is becoming of increasing value is high throughput process development (HTPD), where the roles of pH and the ionic strength are very important as they rule binding behaviour of target(s) and contaminants(s). By successful design of such high throughput processes, the conditions for high mass transfer rates and accordingly process economy, and also for optimal elution and hence highest recovery, can be accurately predicted. However, such successful design would require or at least be much improved by automatic buffer preparation, allowing the preparation of numerous conditions such as pH and I in short time spans.
Another need of precise and well controlled buffer preparation appears in microplates and other labware formats. Many steps in microplate and filter plate based assays are easily parallelized by using e.g. multi-pipettes and vacuum blocks for the processing and plate readers for the detection of results. There is often no need for higher level of automation, i.e. automated transportation of plates. This is valid even for many “throughput applications” since a factor 96 or 384 in the number of experiments is already gained by the plate integration in the first place. However, there are cases involving tedious preparation of individual wells in the plate. While dedicated, small footprint plate readers are taken for granted for the detection and analysis of individual wells there is no similar dedicated, reasonably priced, small footprint solution for buffer preparation in individual wells in one microplate. TECAN is a company which has addressed the problem by writing software to provide buffer preparation in microplates using their lab automation platform. This could at a first glance be perceived as an elegant solution, but results in occupation of an expensive automation infrastructure for hours for a relatively simple task in the well equipped automation lab. Furthermore, a huge investment is required for the small lab to take advantage from the TECAN solution.
Thus, there is also a need of an automatically dispensing device, such as a stand-alone unit, which may be used as a workstation together with automation solutions e.g. to reduce the workload in a primary automation infrastructure.